Question
What does it mean for the metric to be scale invariant in curved spacetime (in the sense when I say a property is scale invariant in thermodynamics)? I'm confused as to how to define this. It seems to be either by means of a Weyl scaling or conformal transformation where the scaling factor is a constant? I suspect the correct way would be via means of coordinate transformations? Is there some nice mathematical condition such a metric would satisfy?
Motivation
Consider the stress energy tensor for a perfect fluid:
$$T^{\mu \nu} = \left(\rho + \frac{p}{c^2} \right) U^{\mu} U^\nu + p g^{\mu \nu}, $$
Now keeping our notation ambiguous:
$$g^{\mu \nu} \to \lambda^2 g^{\mu \nu}$$
But $$g^{\mu \nu} g_{\mu \nu} = 4$$
Thus
$$ g_{\mu \nu} \to \frac{1}{\lambda^2}g_{\mu \nu} $$
We also know:
$$ g_{\mu \nu} U^{\mu} U^\nu = c^2 $$
Thus,
$$ U^{\mu} U^\nu \to \lambda^2 U^{\mu} U^\nu $$
Thus we have effectively done the following:
$$ T^{\mu \nu} \to \lambda^2 T^{\mu \nu} $$